Temperature distributions with more than one independent variable

1

Temperature Distributions With More

Than One Independent Variable

By:

Ihsan Ali Wassan

(14CH18)

Chemical Engineering Department

Quaid-e-Awam University of Engineering Science & Technology, Nawabshah, Sindh, Pakistan

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE

Presentation Outlines

Temperature Distribution?

Temperature Distribution over Time Graph

Steady Vs Unsteady Heat Conduction

Unsteady Heat Conduction in Solids

Heating of a Semi-Infinite Body Or Slab

2TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE

Fourier's law of heat conduction, which gives a first-order differential equation

for the temperature as a function of position.

• Fourier's law allows us to determine temperature distribution in a medium.

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 3

Temperature Distribution?


Temperature Distribution over Time Graph

Steady Vs Unsteady Heat Conduction

Steady implies no change with

time at any point within the

medium.


Unsteady (Transient) implies

variation with time or time

dependence.


General Equation

If thermal conductivity can be assumed to be independent of the temperature & position, then

where Thermal diffusivity of solid.



Four important methods for solving unsteady heat conduction problems:

1.the method of combination of variables,

2.the method of separation of variables,

3.the method of sinusoidal response, and

4.the method of Laplace transform.


1. Method of combination of variables (or the method of similarity solutions):

This method is useful only for semi-infinite regions, such that the initial

condition and the boundary condition at infinity may be combined into a single

new boundary condition.

2. Method of separation of variables:

The partial differential equation is split up into two or more ordinary

differential equations.



3. Method of sinusoidal response:

Useful in describing the way a system responds to external periodic disturbances.

4. Method of Laplace transform:

By applying the Laplace transform, we can change an ordinary differential equation

into an algebraic equation.



Heating of a Semi-Infinite Body Or Slab


1. The Microscopic Energy Balance in the y direction states that

2. Dimensionless Variable

simplify


-------------------(a)

-------------------(b)

Solution

3. The boundary and initial conditions states that:


Solution

4. since is dimensionless variable, it must be related to

Therefore

Where

This is the "method of combination of (independent) variables."


Solution


5. The differential equation (b) can be broken down from a PDE to ODE.

(a) First Taking L.H.S

Multiply and divide by

-----------------(b)

Solution

The Value for can be found from taking the derivative of with respect to .

This yields


--------------(c)

Solution

t

(b) Now taking R.H.S

The value for can be found from taking derivative of with respect to .

This yields


Solution

We want so,

Put Eq. (c) and (d) in Eq. (b)

We get


Solution

-------------------(d)

------------------(e)

This is an ordinary differential equation


Solution

Temperature Profile


Temperature Distribution in Dimensionless form


Solution


Solution


Solution

23

Thank You

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE